The surpisingly small size of the Beis Hamikdash and it's effect on Korban Pesach

Since we are in the middle of the 9 days, mourning the destruction of the Beis Hamikdash, this is an appropriate time to discuss issues relating to the Beis Hamikdash.

The Azara of the Beis Hamikdash was surpisingly small, 187 x 135 amos, 25425 square amos. Even assuming a large amah like the Chazon Ish, of 24 inches, that makes the Azara about 50,000 square feet. Of course the Azara had things in it like the kevesh, mizbeach, kiyor, shulchanos, amudim and taba'os which took up space, but even ignoring these this would allow a maximum of 10,000 people (5sq feet a person) in the Azara at one time. Even halving the space requirements per person to 2.5 sq feet a person gets us to a maximum of 20,000 people in the Azara at one time (again ignoring all of the "furniture").

This becomes very important on Erev Pesach when the Korban Pesach needs to be sacrificed in the afternoon. The Gemara in Pesachim (64b) states that King Agrippas wanted to take a census so he had the Kohanim count the number of Korban Pesachs that were brought and it came out to 1.2 million korbanos, 12 million people (10 to a korban).

How could they possibly have fit all of the people and animals in the Azara? The Gemara states that the Korban Pesach is brought in 3 shifts. 1.2 million korbanos means 1.2 million animals and at least 1.2 million people, even in 3 even shifts that is 400,000 people and animals a shift!!! As was pointed out above there is no way to fit in even a fraction of the people and animals required per shift. In addition, moving 400,000 people into and out of a confined space takes hours, the whole 3.5 hours would have been taken up by simply trying to get the people in and out without having time to do anything.

This leads to the question that given the dimensions of the Beis Hamikdash and the number of people, how did everyone ever bring a korban pesach? The numbers just don't work.

There is another related question that needs to be raised as well. The Korban Pesach is only brought in the afternoon of the 14th of Nisan after the korban tamid is brought. The Mishna (58a) states that on Erev Pesach the Tamid was brought from 7.5 - 8.5. That leaves just 3.5 hours (8.5 -12) to bring the korban pesachs. That means that they had to sacrifice over 340,000 korbanos an hour, over 5700 a minute and over 95 a second. That is simply impossible. Let's think about what needed to be
done.
A. shechita
B. collect the blood
C. sprinkle the blood on the mizbeach
D. Skin the animal and take out the organs to be burned on the mizbeach.

All of this at the pace of 95 a second???

Just for comparison a modern slaughterhouse kills about 1000 animals an
hour, so in the time allotted could get to 3500 animals.

Nowhere in the Gemara (or in any of the Rishonim/Acharonim that I saw) is
there even a hint that these numbers are not real.

Some want to answer that you are right, it's impossible under normal conditions, and therefore, it must have been miraculous. However, this answer is very difficult for a number of reasons:

1. The Gemara on the same Daf (Pesachim 64b) has a dispute between Abaye and Rava how the doors to the Azara were closed when they were makriv the korban pesach. Abaye says that they were closed by a miracle and Rava says that they were closed by the Kohanim and the dispute revolves around whether we can rely on a miracle to close the doors. However, if the entirety of the hakrava of the korban pesach was ma'aseh nissim from beginning to end, then Rava's position is completely untenable.  Nobody went into the Azarah for pesach without relying on countless miracles, so why is the closing of a door by a miracle a problem but all of the other miracles were not? Additionally, in a machlokes Abaye and Rava we always pasken like Rava (except for 6 cases, this not being one of them) so halacha l'ma'seh we should pasken that we can't rely on miracles.
2. There is no source in Chazal for saying that bringing the Korban Pesach required multiple major miracles. For example, Chazal do state that there was a miracle on Yom Kippur that the people were standing packed together and yet had room to bow. The miracles required for bringing the Korban Pesach are much bigger and greater then this and yet there is no mention. 
3. We have a general principle of אין סומכים על הנס, it would be a gross violation of this principle to have a very important mitzva (if you don't do it you are חייב כרת) that can only ever be done with miracles. 

All of the questions that we asked above about the Korban Pesach in the Beis Hamikdash are even more difficult regarding the Korban Pesach that was brought in the Midbar in the second year.  The Mishkan was much smaller then the Beis Hamikdash and there were only 3 Kohanim at that time in the Midbar, Aharon and his 2 sons. So in addition to the questions already asked above, we can ask another question. How could 3 Kohanim possible have done all of the Avoda for hundreds of thousands if not million of korbanos?

Based on all of the above we have the following choices to explain Korban Pesach:

1. The masses of Jews never actually came to the Beis Hamikdash and brought the Korban Pesach
2. Ignore all of the above and rely on miracles to believe that million of Korban Pesachs were brought in 3.5 hours.